Let $n \in \mathbb{N}$, let $U \subset \mathbb{R}^n$ be open, and let $x_0 \in U$. Let $f: U \to \mathbb{R}^n$ and $g: U \to \mathbb{R}^n$ be smooth vector fields, and let $h: U \to \mathbb{R}$ be a smooth output map. Consider the single-input control-affine system
For a smooth function $\varphi: U \to \mathbb{R}$, write $L_f\varphi(x)=D\varphi_x(f(x))$ and $L_g\varphi(x)=D\varphi_x(g(x))$. Assume that the system has relative degree $r$ at $x_0$, meaning that after possibly replacing $U$ by a neighbourhood of $x_0$,
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\begin{align*}
L_gL_f^k h(x)=0 \quad \text{for every } x \in U \text{ and every } k \in \{0,\dots,r-2\},
\end{align*}
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and
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\begin{align*}
L_gL_f^{r-1}h(x) \neq 0 \quad \text{for every } x \in U.
\end{align*}
are linearly independent for every $x$ in a neighbourhood of $x_0$, and that there exist smooth functions $\psi_1,\dots,\psi_{n-r}:U \to \mathbb{R}$ such that
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\begin{align*}
L_g\psi_j(x)=0 \quad \text{for every } x \in U \text{ and every } j \in \{1,\dots,n-r\},
\end{align*}
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and the map
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\begin{align*}
\Phi: U \to \mathbb{R}^n, \quad x \mapsto \bigl(h(x),L_fh(x),\dots,L_f^{r-1}h(x),\psi_1(x),\dots,\psi_{n-r}(x)\bigr)
\end{align*}
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has invertible derivative $D\Phi_{x_0}:\mathbb{R}^n \to \mathbb{R}^n$.
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Then, after restricting to a smaller neighbourhood $V \subset U$ of $x_0$, $\Phi$ is a coordinate chart on $V$. In the coordinates